Chapter 9.3, Problem 29E

Let Y₁, Y₂, … , Y_n denote a random sample of size n from a power family distribution (see Exercise 6.17). Then the methods of Section 6.7 imply that Y_(n) = max(Y₁, Y₂, … , Y_n) has the distribution function given by

$F_{(n)}=\{\begin{array}{ll}0,\hfill & y<0,\hfill \\ {(y/\theta )}^{\alpha n},\hfill & 0\le y\le \theta ,\hfill \\ 1,\hfill & y>\theta .\hfill \end{array}$

Use the method described in Exercise 9.26 to show that Y_(n) is a consistent estimator of θ.